Orthogonal A-Trails of 4-Regular Graphs Embedded in Surfaces of Low Genus

Anton Kotzig has shown that every connected 4-regular plane graph has an A-trail, that is an Euler trail in which any two consecutive edges lie on a common face boundary. We shall characterise the 4-regular plane graphs which contain twoorthogonalA-trails, that is to say two A-trails for which no subtrail of length 2 appears in both A-trails. Our proof gives rise to a polynomial algorithm for deciding if two such A-trails exists. We shall also discuss the corresponding problem for graphs in the projective plane and the torus, and the related problem of deciding when a 2-regular digraph contains two orthogonal Euler trails.